Ok - many bits to this, and dont want to be unduly nitpicky but there are some real misperceptions of therapy and it is helpful to be clear about things when you are looking for a new therapist.
Most psychotherapy is talk therapy to be honest, including CBT - EMDR is an exception to this. Creative therapy, drama therapy, dance therapy, equine assisted therapy are non talk therapies. But if you are sitting talking to someone in a room, it is talk therapy. I seem lots of people - even health professionals and GPs use this incorrectly.
Some types of talk therapy come from the psychoanalytic tradition, where the therapists job is to listen to you without providing direction or advice. Other types will incorporate more problem solving or practical approaches. Its a good idea to ask someone what primary modality they use, the theoretical approach that informs their work, and what their expectations are of you in therapy.
Lots of people say they do CBT when what they mean is they use some worksheets and know what CBT means. Some people use CBT but learning how to do this is their only experience of doing therapy (lots of IAPT staff are retrained nurses, OTs or other people who have top up training in delivering CBT). IME, while some of these are good they are not really psychotherapists and are limited in what they can do and offer. That may suit what you need, in which case great. If not, there are CBT therapists who are primarily psychotherapists (their training and background is providing therapy, and they have usually come through psychology training routes) - they will often have experience in using several types of therapy, but CBT underpins their approach. They can help with a wider range of problems than the other kind of CBT therapist.
I would look for someone who is accredited through the BABCP. They have a register of therapists.
Take care.